few_shots: Example:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS
options: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
problem_activity: Activity
In this activity, students calculate the surface area to volume ratios of animals with different shapes and think about how this ratio might be important in biology. Note that this is the first time students will be using formulas for the surface area and volume of a sphere, and that these formulas are given without derivation. Students used the sphere volume formula in grade 8.
Launch
“Assume a spherical cow” is a common joke about physics. Sometimes in physics, everything about the world is made as simple as possible. There is no air resistance, and everything is modeled with ideal shapes like a sphere or a point. Tell students they will do the same today with spherical elephants and cylindrical snakes.
Student Facing
For a sphere with radius
\(r\)
, its volume is
\(\frac43 \pi r^3\)
and its surface area is
\(4\pi r^2\)
.
Let’s model an elephant with a sphere that has a radius of 4.5 feet.
What is the volume of the elephant?
What is the surface area of the elephant?
Let’s model a snake with a cylinder of length 3 feet and diameter 0.2 feet.
What is the volume of the snake?
What is the surface area of the snake?
Compute the surface area to volume ratio, or
\(\frac{SA}{V}\)
, for each animal.
Student Response
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Activity Synthesis
Ask students how the surface area to volume ratio could affect the biology of snakes and elephants. The goal is for students to notice that the snake has a much higher ratio and to think about the impacts this may have.
One aspect of the biology is that elephants are warm-blooded, so to maintain a consistent body temperature, their bodies need to minimize the surface area through which heat enters or escapes their bodies. Snakes are cold-blooded, so their bodies maximize surface area in order to absorb heat from the environment through their skin.


correct_option_index: 0